### Practice Problem Set 8 – Expected Insurance Payment – Additional Problems

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This practice problem set is to reinforce the 3-part discussion on insurance payment models (Part 1, Part 2 and Part 3). The practice problems in this post are additional practice problems in addition to Practice Problem Set 7. Another problem set on expected insurance payment: Practice Problem Set 9.

 Practice Problem 8A Losses follow a uniform distribution on the interval $(0,50000)$. An insurance policy has an 80% coinsurance and an ordinary deductible of 10,000. The coinsurance is applied after the deductible so that a positive payment is made on the loss amount above 10,000. Determine the expected payment per loss.
 Practice Problem 8B Losses follow a uniform distribution on the interval $(0,50000)$. An insurance policy has an 80% coinsurance and an ordinary deductible of 10,000. The coinsurance is applied after the deductible so that a positive payment is made on the loss amount above 10,000. In addition to the deductible and coinsurance, the coverage has a policy limit of 24,000 (i.e. the maximum covered loss is 40,000). Determine the expected payment per loss.
 Practice Problem 8C Losses in the current year follow a uniform distribution on the interval $(0,50000)$. Further suppose that inflation of 25% impacts all losses uniformly from the current year to the next year. Losses in the next year are paid according to the following provisions: Coverage has an ordinary deductible of 10,000. Coverage has an 80% coinsurance. The coinsurance is applied after the deductible. The coverage has a policy limit of 24,000. Determine the expected payment per loss.
 Practice Problem 8D Liability claim sizes follow a Pareto distribution with shape parameter $\alpha=1.2$ and scale parameter $\theta=10000$. Suppose that the insurance coverage has a franchise deductible of 20,000 per loss. Given that a loss exceeds the deductible, determine the expected insurance payment.
 Practice Problem 8E Losses in the current year follow a Pareto distribution with parameters $\alpha=3$ and $\theta=5000$. Inflation of 10% is expected to impact these losses in the next year. The coverage for next year’s losses has an ordinary deductible of 1,000. Determine the expected amount per loss in the next year that will be paid by the insurance coverage.
 Practice Problem 8F Losses in the current year follow a Pareto distribution with parameters $\alpha=3$ and $\theta=5000$. Inflation of 10% is expected to impact these losses in the next year. The coverage for next year’s losses has a franchise deductible of 1,000. Determine the expected amount per loss in the next year that will be paid by the insurance coverage.
 Practice Problem 8G Losses follow a distribution that is a mixture of two equally weighted exponential distributions, one with mean 6 and the other with mean 12. An insurance coverage for these losses has an ordinary deductible of 2. Calculate the expected payment per loss.
 Practice Problem 8H Losses follow a distribution that is a mixture of two equally weighted exponential distributions, one with mean 6 and the other with mean 12. An insurance coverage for these losses has a franchise deductible of 2. Calculate the expected payment per loss.
 Practice Problem 8I You are given the following information. Losses follow a distribution with the following cumulative distribution function. $\displaystyle F(x)=1-\frac{1}{3} e^{-2x}-\frac{1}{3} e^{-x}-\frac{1}{3} e^{-x/2} \ \ \ \ x>0$ For each loss, the insurance coverage pays 80% of the portion of the loss that exceeds a deductible of 1. Determine the average payment per loss.
 Practice Problem 8J You are given the following information. Losses follow a lognormal distribution with $\mu=3$ and $\sigma=1.2$. An insurance coverage has a deductible of 10. Determine the percentage change in the expected claim cost per loss when losses are uniformly impacted by a 20% inflation.

All normal probabilities are obtained by using the normal distribution table found here. $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

8A
• 12,800
8B
• 12,000
8C
• 14,400
8D
• 170,000
8E
• 1968.9349
8F
• 2574.761038
8G
• $\displaystyle 3 e^{-1/3}+6 e^{-1/6}=7.2285$
8H
• $\displaystyle 4 e^{-1/3}+7 e^{-1/6}=8.7915$
8I
• $\displaystyle 0.8 \biggl(\frac{1}{6} e^{-2}+\frac{1}{3} e^{-1}+\frac{2}{3} e^{-1/2} \biggr)=0.43963$
8J
• Claim Cost before inflation: 32.52697933.
• Claim Cost after inflation: 40.51721002.
• 24.56% change.

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